The angle of depression is the angle between the horizontal line of sight and the line of sight down to an object. It is therefore angle C.

Using right-angle triangle trigonometry, we can find the angle of elevation:

\(cos(\theta)=\frac{10}{15}\)

So, \(\theta=\cos^{-1}\left(\frac{10}{15}\right)\)

\(\theta=48.2^{o}\)

First finding the height of the right-angled triangle:

\(\tan\left(43^{o}\right)=\frac{h}{10}\)

\(h=10\cdot\tan\left(43^{o}\right)\)

\(h=9.3251...m\)

Given that the person is half the height of the tree, we can now double the previous answer to obtain the height of the tree.

Height of Tree \(=18.7 \space m\)

Here is the worded statement translated into a picture:

Using right-triangle trigonometry: SOH CAH TOA

\(\tan\left(87\right)=\frac{x}{20}\)

So \(x=20\cdot\tan\left(87\right)\)

\(x= 382 \space{m} (3sf)\)

A bearing is a 3-digit angle which is measured clockwise from North. i.e. you start at the North line and identify the angle turned. Therefore, in this case the bearing is \(030^{o}\).

A bearing is a 3-digit angle which is measured clockwise from North. i.e. you start at the North line and identify the angle turned. Therefore, in this case the bearing is \(180^{o}+110^{o}=290^{o}\).

By drawing a sketch, we first see that finding the supplementary angle (\(100^{o}\)) then allows us to calculate the bearing of A from B (or B to A).

Starting with a sketch:

Once the sketch has been drawn, use the cosine rule to find the missing length.

\(x^{2}=60^{2}+45^{2}-2\cdot60\cdot45\cdot\cos\left(65\right)\)

\(x = 57.8 \space{km}\)

The bearings in the correct sketch both turn clockwise from North, whereas the second bearing in the incorrect skecth turns anti-clockwise.

Start by finding the included angle between 200 m and 160 m. This is \(360^{o}-90^{o}-162^{o}=108^{o}\). Then use the cosine rule to find the missing length:

\(x^{2}=200^{2}+160^{2}-2\cdot200\cdot160\cdot\cos\left(108\right)\)

\(x = 292 \space{m}\)